196 Sir William R. Hamilton on the Argument of Abel, 



« (m) » ^ » 





which would not involve the radical a ; so that in this way the number of 



extractions of prime roots of variables might be diminished, which would be 



W 

 inconsistent with the irreducibllity of 6 . - 



The results of the present article may be exemplified in the case of any one 

 of the functions V, b", b'", V, which have already been considered. Thus, in 

 the case of the function 6", which represents a root of the general cubic equa- 

 tion, we have 



T" = t", c"-\,r' = f',b'' -b' ,U' z=t" .tl'-^", 

 III II /3." /3," /3," /3," 



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and the n -\- t =14-3 = 4 followmg relations hold good : 



fin — fi fii — II \ —h' fit "-2 — A ' . 



of which indeed the third is identically true, and the second does not involve o,', 



because bj = — -5- ; but both the first and fourth of these relations involve that 



c —a' 

 radical o,', because// = c, -f- o/, and b.^ = - '^a . 



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[10.] Since each of the t terms of the development of i can be expressed 

 as a rational function of the s roots »,, . . . a;, of that equation of the s"" degree 



which b is supposed to satisfy ; it follows that every rational function of these 



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t terms must be likewise a rational function of those s roots, and must admit, 

 as such, of some finite number r of values, corresponding to all possible changes 

 of arrangement of the same s roots among themselves. The same term or 

 function must, for the same reason, be itself a root of an equation of the r"" 

 degree, of which the coefficients are symmetrical functions of the s roots, 

 a;, , . . . X,, and therefore are rational functions of the s coefficient a,, . . . a,, and 

 ultimately of the n original quantities a,, . . . a„; while the r — 1 other roots of 

 this new equation are the r — 1 other values of the same function of a;, , . . . x, , 

 corresponding to the changes of arrangement just now mentioned. Hence, 



